Chapter 2 Portfolio Optimization

Please go to the following link for new version

https://bookdown.org/shenjian0824/portr/

2.1 Prerequisites

Some basic knowledge about finance, time series analysis, optimization(linear and convex) would be preferred. But this chapter we will mainly focus on optimization problems from finance and using python/R to solve them.

2.2 Foundations

2.2.1 Asset log-prices

The fundamental asset pricing model is based on modeling log-price as a random walk:

$$y_{t}\triangleq\log p_{t}$$ $$y_{t}=\mu+y_{t-1}+\epsilon_{t}$$

Comment:

• This assumes log-return is “stationary”
• One may use some statistical test for this assumption. (google it)
• Few interesting observations should be noticed when modeling
  • Autocorrelation
  • Non-Gaussianity and asymmetry with heavy-tailness
  • Volatility clustering
    
  • Conditional heavy-tailness

2.2.2 Return

2.2.2.1 Overview

Our objective: modeling $r_t$ conditional on $\mathcal{F_{t-1}}$

Overall assumption: $r_t$ is multivariate stochastic process with conditional mean and covariance matrix denoted as $$\begin{aligned} \boldsymbol{\mu}_{t} &\triangleq\textsf{E}\left[\mathbf{r}_{t}\mid\mathcal{F}_{t-1}\right]\\ \boldsymbol{\Sigma}_{t} &\triangleq\textsf{Cov}\left[\mathbf{r}_{t}\mid\mathcal{F}_{t-1}\right]=\textsf{E}\left[(\mathbf{r}_{t}-\boldsymbol{\mu}_{t})(\mathbf{r}_{t}-\boldsymbol{\mu}_{t})^{T}\mid\mathcal{F}_{t-1}\right]. \end{aligned}$$

Markowitz 1952 model, simplest case:

1. $r_t$: i.i.d,
2. both the conditional mean and conditional covariance are constant $$\begin{aligned} \boldsymbol{\mu}_{t} &= \boldsymbol{\mu},\\ \boldsymbol{\Sigma}_{t} &= \boldsymbol{\Sigma}. \end{aligned}$$

Factor model:

1. $r_t$: i.i.d,
2. constant mean,
3. covariance matrix could be decomposed into two parts: low dimenstional factors and marginal noise(see below),

$$\mathbf{r}_{t}=\boldsymbol{\alpha}+\mathbf{B}\mathbf{f}_{t}+\mathbf{w}_{t}$$ where

• $\boldsymbol{\alpha}$ denotes a constant vector
• $\mathbf{f}_{t}\in\mathbb{R}^{K}$ with $K\ll N$ is a vector of a few factors that are responsible for most of the randomness in the market
• $\mathbf{B}\in\mathbb{R}^{N\times K}$ denotes how the low dimensional factors affect the higher dimensional market assets
• $\mathbf{w}_{t}$ is a white noise

Time-series models:

• To capture time correlation(ACF) we have mean models: VAR, VARIMA
• To capture the volatility clustering we have covairiance models: ARCH, GARCH

2.2.2.2 Estimating

Estimating methods

• sample mean and sample covariance matrix
• many more sophisticated estimators, shrinkage, Black-Litterman

Estimator

• Least-square(LS) estimator
• MLE

Comment

These methods would be only good when data set is large. But we are facing a delimma when data sample could not be large enough due to:

• unavailability
• or lack of stationarity

As a consequence, the estimates contain too much estimation error(ref Markowitz model)